If you can solve this I will give you a million dollars

Is this an AoPS question?

oops this was wrong

Wait you wont give the person who solves this one million dollars i bet :p

I'll give you the answer but you have to solve it.

∠BAP = ∠ACB = ∠RAC = arctan( AB / AC ) 

Assuming that P, Q, and R are on BC (in the order B, P, Q, R, C):

Angle(BAC) = 90.

Since AQ is an angle bisector, angle(BAQ) = 45.

Since P is between B and Q and angle(PAQ) = 13, angle(BAP) = angle(BAQ) - angle(PAQ)

              --->   angle(BAP) = 45 - 13 = 32.

Since AP is an altitude, angle(BPA) = 90.

In triangle(BAP), angle(BAP) = 32 and angle(BPA) = 90   --->   angle(ABP) = 58.

In triangle(BAC), angle(BAC) = 90 and angle(ABP) = 58   --->   angle(ACB) = 32.

Since AR is a median of a right triangle, RA = RB = RC.

Since RA = RC, triangle(ARC) is an isosceles triangle.

Since angle(ACB) = 32, ange(CAR) = 32.